"I hope they're working day and night to get this done. And when they get it fixed, I hope they fire some people that were in charge of making sure that this thing was supposed to work."

Gibbs, an MSNBC contributor, called the problems with Healthcare.gov "excruciatingly embarrassing" for the White House and the Department of Health and Human Services.

"This was bungled badly," he said. "This was not a server problem … just too many people came to the website. This is a website architecture problem. I think it is, again, excruciatingly embarrassing. It's not fatal because there are still many weeks and days to go before the enrollment period closes at the end of March."

The Obama administration initially blamed Healthcare.gov's problems on the crush of traffic on the site, but has since acknowledged the site also has structural flaws – and Republican lawmakers last week demanded answers to the problems.

And Sen. Pat Roberts, R-Kan., and Rep. John Fleming, R-La., have said HHS Secretary Kathleen Sebelius should resign over the site's problems, The Hill reported.

Healthcare.gov – the primary portal to enroll in coverage in 36 states – still blocks some users from creating accounts, which prevents them from even comparing their options for insurance, The Hill noted.